The proof connecting the QFI to GME is for operators linear in the spin degrees of freedom [1]}. This is advantageous from the perspective of response function experiments, since in a typical scattering experiment one imagines that that scattering beam is exploring the response of the state to an applied operator that is often linear in the local degrees of freedom. From a theoretical perspective, however, we can explore parameterizations corresponding to a wide range of operators which would not necessarily be linear in the local degrees of freedom and hence have no established connection to the GME.
For instance, the fidelity susceptibility (FS) [2]}, [3]}, [4]}, [5]}
is identical to the QFI up to a constant of proportionality [6]}. In this context one considers the overlap in the ground state of a Hamiltonian depending on a parameter \(\lambda \)  with the ground state at \(\lambda + \lambda \) , where the parameter coupled to \(\lambda \)  might be non-local in the spin degrees of freedom. Thus, the FS is not generally known to have a relationship with the GME. An analogous relationship to response functions holds true of the FS [7]}.
